The 13th OpenFOAM Workshop (OFW13), June 24-29, 2018, Shanghai, China

DEVELOPMENT OF A MULTIPHASE SOLVER FOR CAVITATION FLOW NEAR FREE SURFACE

HOUCUN ZHOU1, MIN XIANG2, SHIWEI ZHAO3, WEIHUA ZHANG4

1National University of Defense Technology, zhouhoucun09@nudt.edu.cn 2National University of Defense Technology, xiangmin333@hotmail.com 3National University of Defense Technology,

15211122522@163.com 4National University of Defense Technology, zhangweihua@nudt.edu.cn

Keywords: Free Surface, Cavitation, Multipase, OpenFOAM

1 IntroductionCavitation is a common phenomenon in fluid machinery, which oftentimes negatively affects the performance of mostfluid machinery applications, therefore leading to problems such as undue vibrations, noise and material erosion [1]. Inmore recent years, cavitation has attracted intensive attention due to its potentials in drag reduction for underwater vehicles[2]. When vehicles run near or across the free surface, ventilated cavitation happens, which is complicated issue and mayprovide new inspiration on high-speed surface cruising. Researchers have done a lot of investigation on this topic up tonow[3][4][5]. In order to fully studies this problem, a multiphase cavitation solver is developed based on the OpenFOAMopen source platform. And a fully investigation of the free surface flow will be carried out with the developed solver.

2 Mathematical Method

2.1 Governing Equation

In this paper, the flow described is treated as a homogeneous mixture, therefore only one set of equations is needed. Thegoverning equations basically consist of the conservation of mass, momentum. The continuity equation of the mixtureflow can be written as

∂ρ

∂t+∇ · (ρU) = 0 (1)

Neglecting the gravity and surface tension term, the conservation of momentum for the mixture flow can be expressed as,

∂ (ρU)

∂t+∇ · (ρU ⊗ U) = −∇p+∇ · τ (2)

Where ρ is the density of the mixture, which is related to the volume of fractions of all phases as

ρ = αlρl + αvρv + αgρg (3)

For a three phases system, the basic form of transport equations for the volume fraction of each phase could write as:

∂ (ρlαl)

∂t+∇ · (ρlαlU) = m (4)

∂ (ρvαv)

∂t+∇ · (ρvαvU) = −m (5)

∂ (ρgαg)

∂t+∇ · (ρgαgU) = 0 (6)

Where the subscript l and v are for the liquid and vapour phases respectively. While g represents gas. Note that thevelocity vector U in the above equations should be expressed as the averaged velocity, and U = αlUl + αvUv + αgUg.And the m term on the RHS of the equations is donates the mass transfer rate caused by cavitation between the liquid andvapour phase, which is m = m+ + m−.Considering that the volume fraction of these phases obey the conservation law,

αl + αv + αg = 1 (7)

201

The 13th OpenFOAM Workshop (OFW13), June 24-29, 2018, Shanghai, China

The divergence of the velocity can be,

∇ · U =

(1

ρl− 1

ρv

)m (8)

Neglecting the compressibility of the phases, and take the divergence term into consideration, the final form of thetransport equation is:

∂αl

∂t+∇ · (αlU) = αl (∇ · U) +

(1

ρl− αl

(1

ρl− 1

ρv

))m (9)

∂αv

∂t+∇ · (αvU) = αv (∇ · U)−

(1

ρv+ αv

(1

ρl− 1

ρv

))m (10)

∂αg

∂t+∇ · (αgU) = αg (∇ · U)− αg

(1

ρl− 1

ρv

)m (11)

Given that the interface disperses due to numerical diffusion it might appear that a good approach to limit or reverse thiseffect would be to include a diffusion operator into the phase-fraction equation with a negative diffusion coefficient. Whilethis approach would be conservative it would also be unbounded and unstable; negative diffusion is always problematic.An alternative to negative diffusion which is also conservative is to apply some kind of additional convection-based termwhich compresses the interface, maintains boundedness and reduces to zero (at least the integrated effect of it reduces tozero) as the mesh is refined. A ”Counter-gradient” term, which is clearly conservative and maintains boundedness, has ageneral form of ∇ · (Urαβ), Uris the relative velocity between the two phases across the interface.

∂αl

∂t+∇ · (αlU) + +∇ · (αlαv (Ul − Uv) + αlαg (Ul − Ug)) = αl (∇ · U) +

(1

ρl− αl

(1

ρl− 1

ρv

))m (12)

∂αv

∂t+∇ · (αvU) +∇ · (αvαl (Uv − Ul) + αvαg (Uv − Ug)) = αv (∇ · U)−

(1

ρv+ αv

(1

ρl− 1

ρv

))m (13)

∂αg

∂t+∇ · (αgU) +∇ · (αgαl (Ug − Ul) + αgαv (Ug − Uv)) = αg (∇ · U)− αg

(1

ρl− 1

ρv

)m (14)

However, in this study, the homogeneous multiphase model which adopts an incompressible Navier-Stokes equations andthe compressive volume of fluid (VOF) approach are adopted for flow simulation. Phases in the multiphase system sharethe velocity and pressure fields. So the relative velocity term should be reconstructed. Here we just adopt the followingform which is also used by some other multiphase solver in OpenFOAM.

Uc = min (cα|U|,max (|U|)) ∇α

∥∇α∥(15)

where cα is a compressive factor, expression max (|U|) returns the largest value of |U | anywhere in the domain.

2.2 Flux Corrected Transport Theory

In OpenFOAM, the volume fraction transport equation of the multiphase system is solved by the MULES tool kit. Forthe given equation below

∂ (ρα)

∂t+∇ · (ραU) = αSp + Su (16)

We have the following discretization equation

ραn+1 − αn

∆t+ F = αn+1Sp + Su (17)

αn+1

(1− ∆tSp

ρ

)=

∆t

ρ

(αnρ

∆t+ Su− F

)(18)

αn+1 =αnρ/∆t+ Su− F

ρ/(∆t)− Sp(19)

Usually, the density is set to unity in MULES correction. Therefore, the Sp term is an implicity volumetric source termdue to cavitation, the Su is an explicit term in Eq.19. And F is the total volume fraction flux due to transportive effectof a velocity. The values of the flux depend on many variables but particularly on the values of α at faces. Boundednessof the temporal solution can be achieved via face value limiting, such as in TVD/NVD schemes, or by limiting the facefluxes. The values of F are obtained by a lower order and bounded method and a limited portion of the values obtained

202

The 13th OpenFOAM Workshop (OFW13), June 24-29, 2018, Shanghai, China

Figure 1: Schematic diagram of limiter for one dimensional geometry.

by a high order and possible unbounded method. This technique is Flux Corrected Transport (FCT). The sequence of thistheory could be described as below[6]:

1. Compute FL, the transportive flux given by some low order scheme which guarantees to give monotonic results.2. Compute FH , the transportive flux given by some high order scheme.3. Define the anti-diffusive flux A = FH − FL.4. Compute the corrected flux FC = FL + λA, with 0 ≤ λ ≤ 1.5. Solve the equation by the given temporal scheme using corrected fluxes.The critical step is clearly the fourth, where it is necessary to find the λ weighting factors.The implementation of FCT theory is called MULES (Multidimensional Universal Limiter for Explicit Solution)[6]

in OpenFOAM. Illustrated by Fig.1, the whole procedure is presented in Algorithm 1.

Algorithm 1 Procedure for MULES limiter1: Calculate local extrema as

αai = max(αn

i , αni,N )

αbi = min(αn

i , αni,N )

where αni,N are all the neighbors by face for the i-th cell. In addition the inflows and outflows for each cell

have to be calculated as P+ = −∑

f A−f and P− = −

∑f A

+f , where A−

f are the inflows and A+f the

outflows;2: Correct the local extrema by the limits imposed by user’s defined global extrema αmaxG and αminG

αai = max(αmaxG, αa

i )αbi = min(αminG, αb

i )

3: Find Q±i as

Q+i = V

∆t(αai − αn

i ) +∑

f FLf

Q−i = V

∆t(αni − αb

i )−∑

f FLf

4: Set λv=1f = 1 for all faces. Do the following loop nLimiterIter times to find the finalλf ’s

λ∓,v+1i = max

[min

(±

∑f λv

fA±f +Q±

i

P±i

, 1

), 0

]λv+1f =

{min{λ+,v+1

P , λ−,v+1N }, ifAi+1/2 ≥ 0,

min{λ−,v+1P , λ+,v+1

N }, ifAi+1/2 < 0

where λP and λN represent the λ’s for the owner and neighbor cell of a given face f .

203

The 13th OpenFOAM Workshop (OFW13), June 24-29, 2018, Shanghai, China

Figure 2: Block diagram of the mass transfer model framework and implemented models.

Figure 3: Sheet cavitation on the suction side of the hydrofoil

2.3 Mass Transfer Model

In order to calculate the mass transfer rate within multiphase systems, a framework of cavitation model is developed.All models inherit from a abstract class, which accepts a dictionary reference to look up parameters needed by the modeland two reference of phase fraction field object. Additionally, the Kunz model[7] and Schnerr Sauer model[8] are alsoimplemented to return the mass transfer rate. A UML block diagram of the mass transfer model framework is presentedin Fig.2.

3 Validation

3.1 2D hydrofoil case

In this section, the two dimensional NACA66(MOD) hydrofoil is adopted. The main purpose of this case is validatingthe performance of two phase flow simulation. So only water and vapour phase are involved in this validation case. Resultscalculated by the new solver will compare with experimental results[9] and those calculated by the interPhaseChangeFoamsolver. Some of the primary results are shown in Fig.3 and Fig.4.

3.2 2D throttle case

This 2D throttle case is copied form cavitatingFoam tutorials.In this case, three phases are involved, i.e. the water,vapour phase and gas. Two different flow condition are considered. Numerical simulation results are shown in Fig.5 andFig.6, which validate the developed solver.

3.3 3D underwater projectile

The effect of the free surface has a great effect on high-speed surface vehicles. Wang et. al [3] carried out atypical launching experiment around an axisymmetric projectile to investigate the free surface effect. In this study, theestablished numerical approach is adopted to study this case. The numerical methods are validated by comparing resultswith underwater launching experiments. Results are shown in Fig.7 and Fig.8.

4 ConclusionThe new developed multiphase cavitation solver shows good performance in two phases and three phases cavitating

flow simulation. It could also be applied to analyse the cavitating flow near free surface.

204

The 13th OpenFOAM Workshop (OFW13), June 24-29, 2018, Shanghai, China

Figure 4: Pressure coefficient distribution on the suction side of the hydrofoil

Figure 5: Volume fraction distribution in the throttle(Pout = 15atm)

Figure 6: Volume fraction distribution in the throttle(Pout = 10atm)

205

The 13th OpenFOAM Workshop (OFW13), June 24-29, 2018, Shanghai, China

Figure 7: Evolutions of the cavity and free surface

Figure 8: Wave profiles on the upper side of the projectile

206

The 13th OpenFOAM Workshop (OFW13), June 24-29, 2018, Shanghai, China

Acknowledgments

The authors gratefully acknowledge support by the National Nature Science Foundation of China (NSFC, Grant NO:51776221). And thank all those involved in the organisation of OFW13 and to all the contributors that will enrich thisevent.

References

[1] J.-P. Franc, J.-P. Franc, J.-M. Michel, and J.-M. Michel, Fundamentals of cavitation, 2004.

[2] B.-K. Ahn, C.-S. Lee, and H.-T. Kim, “Experimental and numerical studies on super-cavitating flow of axisymmetriccavitators,” International Journal of Naval Architecture and Ocean Engineering, vol. 2, no. 1, pp. 39–44, mar 2010.[Online]. Available: http://linkinghub.elsevier.com/retrieve/pii/S2092678216302308

[3] Y. Wang, X. Wu, C. Huang, and X. Wu, “Unsteady characteristics of cloud cavitating flow near the free surfacearound an axisymmetric projectile,” International Journal of Multiphase Flow, vol. 85, pp. 48 – 56, 2016. [Online].Available: http://www.sciencedirect.com/science/article/pii/S0301932215300999

[4] M.-S. Jin, C.-T. Ha, and W.-G. Park, “Numerical study of ventilated cavitating flows with free surface effects,”Journal of Mechanical Science and Technology, vol. 27, no. 12, pp. 3683–3691, Dec 2013. [Online]. Available:https://doi.org/10.1007/s12206-013-0914-0

[5] C. Xu, J. Huang, Y. Wang, X. Wu, C. Huang, and X. Wu, “Supercavitating flow around high-speed underwaterprojectile near free surface induced by air entrainment,” AIP Advances, vol. 8, no. 3, p. 035016, 2018. [Online].Available: https://doi.org/10.1063/1.5017182

[6] S. M. Damian, “An extended mixture model for the simultaneous treatment of short and long scale interfaces,” 2013.

[7] R. F. Kunz, D. A. Boger, T. S. Chyczewski, D. R. Stinebring, H. J. Gibeling, and T. R. Govindan, “Multi-Phase CfdAnalysis of Natural and Ventilated Cavitation About Submerged Bodies,” FEDSM 99 1999 ASME/JSME Joint FluidsEngineering Conference, pp. 1–9, 1999.

[8] G. H. Schnerr and J. Sauer, “Physical and numerical modeling of unsteady cavitation dynamics,” in Proceedings ofthe 4th International Conference on Multiphase Flow, New Orleans, La, USA, 2001.

[9] P. E. Dimotakis, H. F. Gaebler, and H. T. Hamaguchi, “Two Dimentional NACA 66(MOD) Hydrofoil High SpeedWater Tunnel Tests,” GALCIT Report HSWT 1142, Tech. Rep., 1987.

207